Theorem bj-inf2vnlem3 13159

Description: Lemma for bj-inf2vn 13161. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-inf2vnlem3.bd1	|- BOUNDED A
bj-inf2vnlem3.bd2	|- BOUNDED Z

Assertion

Ref	Expression
bj-inf2vnlem3	|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Distinct variable groups:   x, y, A   x, Z, y

Proof of Theorem bj-inf2vnlem3

Dummy variables z t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem2 13158	. . 3 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) )
2		bj-inf2vnlem3.bd1	. . . . . 6 |- BOUNDED A
3	2	bdeli 13033	. . . . 5 |- BOUNDED z e. A
4		bj-inf2vnlem3.bd2	. . . . . 6 |- BOUNDED Z
5	4	bdeli 13033	. . . . 5 |- BOUNDED z e. Z
6	3, 5	ax-bdim 13001	. . . 4 |- BOUNDED ( z e. A -> z e. Z )
7		nfv 1508	. . . 4 |- F/ z ( t e. A -> t e. Z )
8		nfv 1508	. . . 4 |- F/ z ( u e. A -> u e. Z )
9		nfv 1508	. . . 4 |- F/ u ( z e. A -> z e. Z )
10		nfv 1508	. . . 4 |- F/ u ( t e. A -> t e. Z )
11		eleq1 2200	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
12		eleq1 2200	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
13	11, 12	imbi12d 233	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
14	13	biimpd 143	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
15		eleq1 2200	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
16		eleq1 2200	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
17	15, 16	imbi12d 233	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
18	17	biimprd 157	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
19	6, 7, 8, 9, 10, 14, 18	bdsetindis 13156	. . 3 |- ( A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) -> A. z ( z e. A -> z e. Z ) )
20	1, 19	syl6 33	. 2 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. z ( z e. A -> z e. Z ) ) )
21		dfss2 3081	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
22	20, 21	syl6ibr 161	1 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Syntax hints:   -> wi 4   \/ wo 697   A.wal 1329   = wceq 1331   e. wcel 1480   A.wral 2414   E.wrex 2415   C_ wss 3066   (/)c0 3358   suc csuc 4282  BOUNDED wbdc 13027  Ind wind 13113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-bdim 13001  ax-bdsetind 13155

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-suc 4288  df-bdc 13028  df-bj-ind 13114

This theorem is referenced by:  bj-inf2vn  13161